Potential new flashpoints in the Middle East unrest have opened after Israel shot at pro-Palestinian protesters on its borders with Syria and Lebanon, killing at least 13 people and drawing furious condemnation from the Syrian regime.

Protests erupted in the West Bank, Gaza and East Jerusalem as well as on Israel's geopolitically sensitive northern borders, as Palestinians clashed with Israeli troops and police and hundreds were injured.

Demonstrators commemorating Nakba day, marking the 1948 war in which hundreds of thousands of people became refugees after being forced out of their homes, were met with live gunfire, rubber bullets, stun grenades and teargas.

Israel accused Syria of provoking the confrontations to divert attention from internal unrest, and said attempts to breach its borders were a provocation intended to exploit Palestinian nationalism in the wake of regional unrest. An Israeli military spokesman said the protests bore "Iran's fingerprints".

"We hope the calm and quiet will quickly return," said the Israeli prime minister, Binyamin Netanyahu. "But let nobody be misled: we are determined to defend our borders and sovereignty."

The defence minister, Ehud Barak, warned "we are just at the start of this matter and it could be that we'll face far more complex challenges".

Syria condemned Israel's "criminal activities". The foreign ministry called on the international community to hold Israel responsible for the deadly confrontation, Syria's state news agency, Sana, said.

Although Israel had been braced for violent protests, the clashes on its borders were largely unexpected. Israeli politicians, already deeply alarmed about uprisings in neighbouring Arab countries, now face heightened tensions with Syria and Lebanon.

Thousands of Palestinian refugees from Syria marched towards the village of Majdal Shams in the Golan Heights, which Israel captured from Syria in 1967.

According to the Israeli military, "hundreds of Syrian rioters infiltrated the Israeli-Syrian border ... and violently rioted against [Israeli] forces". It said its troops "fired selectively towards rioters". Six people were confirmed killed, but there were reports of up to 10 deaths.

"The Israeli army warned [the protesters] not to cross but they didn't listen," Shefa Abu Jabal, 25, a resident of Majdal Shams, said. "When the crowd started to come over ... soldiers started shooting.

"Around 200 have managed to get across. I've heard there are four people dead on this side and there are many more injured. People in the village are really scared. The Israel soldiers looked shocked. No one thought there would be trouble at this border."

Another resident, Hamad Awidat, said: "There are thousands and thousands of people on the Syrian border who are trying to cross. There has been a lot of fighting, and of course people are scared."

At Maroun al-Ras, on the border with Lebanon, witnesses said Israeli troops had fired at protesters throwing stones from within Lebanon, a move that could have serious repercussions and prompt further cross-border incidents.

At least six people were killed after hundreds of protesters broke through Lebanese army barricades to throw rocks across the border. One man, apparently shot in the chest, was doused with water as protesters tried to revive him but shouts of "Allah Akhbar" broke out as his dead body was lifted over the crowd. One protester, his clothes soaked in blood, screamed: "Murderers, cowards, is a rock any match for a bullet?"

Hezbollah, which controls Lebanon's southern villages, had given tacit support for the protest but the crowd was dispersed by Lebanese troops firing into the air.

Yassir Ali, one of the protest organisers, said the deaths were not unexpected. "Palestinian people are used to paying with their lives. It's a big price, but one we are prepared to pay to prove our right to return to the motherland."

Brigadier General Yoav Mordechai, an Israeli military spokesman, said soldiers fired when demonstrators began vandalising the border fence. The army was aware of casualties, he said.

Confrontations were reported after about 600 people marched from the West Bank's principal city, Ramallah, towards the Qalandia checkpoint into Jerusalem. There were clashes in other areas of the West Bank.

In Gaza, at least 80 people were injured when Israeli troops opened fire on demonstrators approaching the Erez border crossing, Palestinian medical sources said. The Israeli military said it shot dead a man trying to plant a bomb near the border.

In Tel Aviv, an Israeli man was killed and 17 people were injured when a truck ran into vehicles and pedestrians. It was not clear whether it was an accident or a deliberate attack. The truck's 22-year-old Israeli-Arab driver said he lost control of the vehicle due to faulty brakes.

The Israeli authorities had expected trouble on the first Nakba day following the Middle East uprisings and had deployed 10,000 soldiers and police.

• Additional reporting: Phoebe Greenwood

• This article was amended on 17 May 2011. Owing to editing changes the original said that at least two people were killed at Maroun al-Ras after hundreds of protesters broke through Lebanese army barricades to throw rocks across the border. This has been corrected.